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Standard XII

Mathematics

Composite Function

Let I denot...

Question

Let $I$ denote the $3 \times 3$ identity matrix and $P$ be a matrix obtained by rearranging the columns of $I$ . Then

There are six distinct choices for

P

and

d e t (P) = 1

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There are six distinct choices for

P

and

d e t (P) = \pm 1

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There are more than one choices for

P

and some of them are not invertible

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There are more than one choices for

P

and

P^{- 1} = I

in each choice

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Solution

The correct option is C There are six distinct choices for $P$ and $d e t (P) = \pm 1$
$I_{(3 \times 3)} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$3 \to$ Columns can be arranged in $3!$ ways
$\Rightarrow 3! = 6$
$\Rightarrow 6$ ways
If even no. of interchanges $\Rightarrow d e t = 1$
If odd no. of interchanges $\Rightarrow d e t = - 1$

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Let I denote the 3×3 identity matrix and P be a matrix obtained by rearranging the columns of I. Then

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